Tilting theory originates in the representation theory of finite dimensional algebras. Today the subject is of much interest in various areas of mathematics, such as finite and algebraic group theory, commutative and non-commutative algebraic geometry, and algebraic topology. The aim of this book is to present the basic concepts of tilting theory as well as the variety of applications. It contains a collection of key articles, which together form a handbook of the subject, and provide both an introduction and reference for newcomers and experts alike.
Author(s): Lidia Angeleri Hügel, Dieter Happel, Henning Krause
Series: London Mathematical Society Lecture Note Series
Publisher: CUP
Year: 2007
Language: English
Pages: 481
Cover......Page 1
Title Page......Page 4
Copyright......Page 5
Contents ......Page 6
1 Introduction page ......Page 10
2 Basic results of classical tilting theory ......Page 18
References ......Page 21
1 Introduction ......Page 24
2 Notation ......Page 25
3 Representation-finite algebras ......Page 27
4 Critical algebras ......Page 33
5 Tame algebras ......Page 35
References ......Page 37
1 Introduction ......Page 40
2 Tilting modules ......Page 41
3 Tilting functors, spectral sequences and filtrations ......Page 44
4 Applications ......Page 52
5 Edge effects, and the case t ..2 ......Page 55
References ......Page 56
1 Introduction ......Page 58
2 Derived categories ......Page 60
3 Derived functors ......Page 72
4 Tilting and derived equivalences ......Page 75
5 Triangulated categories ......Page 81
6 Morita theory for derived categories ......Page 87
7 Comparison of t-structures, spectral sequences ......Page 92
8 Algebraic triangulated categories and dg algebras ......Page 99
References ......Page 106
6 Hereditary categories ......Page 114
1 Fundamental concepts ......Page 115
2 Examples of hereditary categories ......Page 117
3 Repetitive shape of the derived category ......Page 121
4 Perpendicular categories ......Page 123
5 Exceptional objects ......Page 124
6 Piecewise hereditary algebras and Happel’s theorem ......Page 126
8 Modules over hereditary algebras ......Page 130
9 Spectral properties of hereditary categories ......Page 133
10 Weighted projective lines ......Page 134
11 Quasitilted algebras ......Page 151
References ......Page 152
1 Some background ......Page 156
3 Basics on Fourier-Mukai transforms ......Page 158
4 The reconstruction theorem ......Page 164
5 Curves and surfaces ......Page 168
6 Threefolds and higher dimensional varieties ......Page 175
7 Non-commutative rings in algebraic geometry ......Page 179
References ......Page 182
8 Tilting theory and homologically finite subcategories with applications to quasihereditary algebras ......Page 188
1 The Basic Ingredients ......Page 190
2 The Correspondence Theorem ......Page 200
3 Quasihereditary algebras and their generalizations ......Page 209
4 Generalizations ......Page 216
References ......Page 220
9 Tilting modules for algebraic groups and finite dimensional algebras ......Page 224
1 Quasi-hereditary algebras ......Page 226
2 Coalgebras and Comodules ......Page 229
3 Linear Algebraic Groups ......Page 234
4 Reductive Groups ......Page 237
5 Infinitesimal Methods ......Page 242
6 Some support for tilting modules ......Page 247
7 Invariant theory ......Page 248
8 General Linear Groups ......Page 250
9 Connections with symmetric groups and Hecke algebras ......Page 253
10 Some recent applications to Hecke algebras ......Page 256
References ......Page 263
1 Introduction ......Page 268
2 The partial order of tilting modules ......Page 269
3 The quiver of tilting modules ......Page 270
4 The simplicial complex of tilting modules ......Page 279
References ......Page 284
11 Infinite dimensional tilting modules and cotorsion pairs ......Page 288
1 Cotorsion pairs and approximations of modules ......Page 290
2 Tilting cotorsion pairs ......Page 301
3 Cotilting cotorsion pairs ......Page 307
4 Finite type, duality, and some examples ......Page 313
5 Tilting modules and the finitistic dimension conjectures ......Page 321
References ......Page 325
12 Infinite dimensional tilting modules over finite dimensional algebras ......Page 332
1 Definitions and preliminaries ......Page 333
2 The subcategory correspondence ......Page 336
3 The finitistic dimension conjectures ......Page 341
4 Complements of tilting and cotilting modules ......Page 345
5 Classification of all cotilting modules ......Page 349
References ......Page 350
13 Cotilting dualities ......Page 354
1 Generalized Morita Duality and Finitistic Cotilting Modules ......Page 357
2 Cotilting Modules and Bimodules ......Page 359
3 Weak Morita Duality ......Page 362
4 Pure Injectivity of Cotilting Modules and Reflexivity ......Page 364
References ......Page 365
1 A brief introduction to modular representation theory ......Page 368
2 The abelian defect group conjecture ......Page 369
3 Symmetric algebras ......Page 370
4 Characters and derived equivalence ......Page 375
5 Splendid equivalences ......Page 379
6 Derived equivalence and stable equivalence ......Page 382
7 Lifting stable equivalences ......Page 384
8 Clifford theory ......Page 385
9 Cases for which the Abelian Defect Group Conjecture has been verified ......Page 387
10 Nonabelian defect groups ......Page 392
References ......Page 393
1 Introduction ......Page 402
2 Spectral Algebra ......Page 405
3 Quillen model categories ......Page 408
4 Differential graded algebras ......Page 412
5 Two topologically equivalent DGAs ......Page 415
References ......Page 418
Appendix Some remarks concerning tilting modules and tilted algebras. Origin. Relevance. Future ......Page 422
1 Basic Setting ......Page 423
2 Connections ......Page 432
3 The new cluster tilting approach ......Page 455
References ......Page 479
